Shenggui Zhang

Color degree and monochromatic degree conditions for short properly colored cycles in edge-colored graphs

For an edge-colored graph, its minimum color degree is defined as the minimum number of colors appearing on the edges incident to a vertex and its maximum monochromatic degree is defined as the maximum number of edges incident to a vertex with a same color. A cycle is called properly colored if every two of its adjacent edges have distinct colors. In this article, we first give a minimum color degree condition for the existence of properly colored cycles, then obtain the minimum color degree condition for an edge-colored complete graph to contain properly colored triangles. Afterwards, we characterize the structure of an edge-colored complete bipartite graph without containing properly colored cycles of length 4 and give the minimum color degree and maximum monochromatic degree conditions for an edge-colored complete bipartite graph to contain properly colored cycles of length 4, and those passing through a given vertex or edge, respectively.

Conditions for graphs to be path partition optimal

Abstract The path partition number of a graph is the minimum number of edges we have to add to turn it into a Hamiltonian graph, and the separable degree is the minimum number of edges we have to add to turn it into a 2-connected graph. A graph is called path partition optimal if its path partition number is equal to its separable degree. We study conditions that guarantee path partition optimality. We extend several known results on Hamiltonicity to path partition optimality, in particular results involving degree conditions and induced subgraph conditions.

Families of vector spaces with r-wise L-intersections

Abstract In this paper, we extend the well-known Frankl–Ray-Chaudhuri–Wilson Theorem on vector spaces to r -wise L -intersecting families, in both modular version and non-modular version. We also provide an upper bound on L -intersecting families of vector spaces which improves the Frankl–Ray-Chaudhuri–Wilson Theorem.

Kernels by properly colored paths in arc-colored digraphs

A {\em kernel by properly colored paths} of an arc-colored digraph

A σ₃ type condition for heavy cycles in weighted graphs

D

A σ_3 type condition for heavy cycles in weighted graphs

is a set

The upper bound of the number of cycles in a 2-factor of a line graph

S

Heavy cycles passing through some specified vertices in weighted graphs

of vertices of

Vertex-disjoint properly edge-colored cycles in edge-colored complete graphs

D

Rainbow triangles in arc-colored digraphs.

such that (i) no two vertices of